, where s is the arclength parameter. Prove that. κ = w = 1 κ (θ) + 1. k (θ + π). dγ dθ. = 1 κ T. = N dn dθ. = T, N T = 0, N = T =1 T (θ + π) = T (θ)

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1 Here is a collection of old exam problems: 1. Let β (t) :I R 3 be a regular curve with speed = dβ, where s is the arclength parameter. Prove that κ = d β d β d s. Let β (t) :I R 3 be a regular curve such that its tangent field T (t) is also regular. Let s be the arclength parameter for β and θ the arclength parameter for T. Show that κ = and det T, dt T,d = T, dt T,d = τ κ. 3. Let γ (θ) be an oval parametrized by θ defined by T = (cos θ, sin θ)of constant wih w = N (θ) (γ (θ + π) γ (θ)) Show that: You can use that w = 1 κ (θ) + 1 k (θ + π). = 1 κ T dt = N dn = T, N T = 0, N = T =1 T (θ + π) = T (θ) 4. Let α (s) unit speed curve with κ>0. Let θ be the arclength parameter for T = dα. Show that the curvature satisfies: κ = 5. Prove that if α (s) is an oval (a closed planar curve with positive curvature and no self intersections), then the unit tangent field T is parallel to T at four or more points. 1

2 6. Let β (t) be a regular curve in R 3 with κ>0. Prove that β is planar if and only if the triple product dβ, d β, d3 β Let γ(t) :I R 3 be a regular curve with positive curvature. Show that γ lies in a plane if and only if the torsion vanishes. 8. Let α (s) =(x (s),y(s)) be a planar unit speed curve. Show that the signed curvature can be computed by κ =det[α,α ] 9. Let α (s) be a unit speed curve in R 3 Prove that det [α,α,α ]=κ τ. It is also possible to find formulas for etc. det [α,α,α ] 10. Prove that the concept of a vertex for a planar curve does not depend on the parametrization. 11. Let γ (t) :I R 3 be a regular curve. Prove that d γ d γ d κ = 1. Let γ(t) :I R 3 be a regular curve with positive curvature. Show that the unit tangent T(t) is a regular and that, if θ is an arclength parameter for T, then dt dn db = 1 κ T = N = T + τ κ B = τ κ B 13. Let γ(t) :I R 3 be a regular curve with positive curvature. Show that γ lies in a plane if and only if the torsion vanishes.

3 14. Let γ (s) =σ (u (s),v(s)) be a unit speed curve on a surface S. Prove that dn = II (T,T) T II (T,C) C, where T =,nis the normal to S, and C = n T. 15. Let X, Y T p S be an orthonormal basis for the tangent space at p to the surface S. Prove that the mean and Gauss curvatures can be computed as follows: H = 1 (II (X, X)+II(Y,Y )), K = II(X, X)II(Y,Y ) (II (X, Y )) 16. Let α :(a, b) R 3 be a unit speed curve with κ (s) = 0for all s (a, b). Define σ (s, t) =α (s)+tα (s). Prove that σ defines a parametrization surface as long as t = 0. Compute the first and second fundamental forms and show that the Gauss curvature K vanishes. 17. For a surface of revolution x (t, θ) =(r (t) cos (θ),r(t)sin(θ),z(t)) compute the first and second fundamental forms and the principal curvatures. 18. Let γ be a curve on the unit sphere S. Prove that its normal curvature κ n is constant. 19. Let σ (u, v) be a parametrized surface. Recall that a tangent vector is a principal direction if it is an eigenvector for the Weingarten map. Assume that the principal curvature are different and show that σ σ u and v are the principal directions if and only if F =0=M. 0. Let α (u) be a unit speed curve in the x, y plane R.Showthat σ (u, v) =(α (u),v). yiel a parametrized surface. Compute its first and second fundamental forms and principal curvatures. Compute its Gauss curvature. 1. Show that the equation ax + by + cz = d defines a surface if and only if (a, b, c) = (0, 0, 0). Show that this surface has a parametrization that is Cartesian.. Let γ be a unit speed curve on a surface S with normal N. Define C = N T, T = γ and κ g = dt C, κ n = dt N, τ g = dc N 3

4 Prove that dt dc dn = κ g C + κ n N, = κ g T + τ g N, = κ n T τ g C. 3. Let γ (u) be a regular curve in the x, y plane R.Showthat σ (u, v) =(γ (u),v). yiel a parametrized surface. Compute its first fundamental form and construct a local isometry from a subset of the plane to the surface. 4. For a regular curve γ (u) :I R 3 {(0, 0, 0)} show that σ (u, v) =vγ (u) defines a surface for v>0 provided γ and γ are linearly independent. Compute its first fundamental form. Show that it admits Cartesian coordinates by rewriting the surface as σ (r, θ) =rδ (θ) for a suitable unit speed curve δ (θ). 5. Let σ (z,θ) = 1 z cos θ, 1 z sin θ, z with 1 <z<1 and π < θ<π.showthatσ defines a patch on a surface. What is the surface? 6. Let σ be a coordinate patch such that E =1and F =0. Prove that the u curves are unit speed with acceleration that is perpendicular to the surface. The u curves are given by γ (u) =σ (u, v) where v is fixed. 7. For a surface of revolution σ (t, θ) =(r (t) cos (θ),r(t)sin(θ),z(t)) show that the first fundamental form is given by E F = ṙ +ż 0 F G 0 r and that the longitudes/meridians γ (t) =σ ((t, θ) have acceleration perpendicular to the surface provided that (r (t), 0,z(t)) is unit speed. 8. Find a conformal map from a surface of revolution σ 1 (r, θ) =(r cos θ, r sin θ, z 1 (r)) to a circular cylinder σ (r, θ) = (cos θ, sin θ, z (r)). 9. Reparametrize the curve (r (u),z(u)) so that the new parametrization σ (t, θ) =(r (t) cos (θ),r(t)sin(θ),z(t)) is conformal. 30. Find an equiareal map from a surface of revolution σ 1 (r, θ) =(r cos θ, r sin θ, z 1 (r)) to a circular cylinder σ (r, θ) = (cos θ, sin θ, z (r)). 31. Reparametrize the curve (r (u),z(u)) so that the new parametrization σ (t, θ) =(r (t) cos (θ),r(t)sin(θ),z(t)) is equiareal. 4

5 3. Let σ : U S be a parametrization of part of the unit sphere. Show that the normal σ u σ v is always proportional to σ. 33. Show that a Monge patch z = f (x, y) is equiareal if and only if f is constant. 34. Show that a Monge patch z = f (x, y) is conformal if and only if f is constant. 35. Show that the equation ax + by + cz = d defines a surface if and only if (a, b, c) = (0, 0, 0). Show that this surface has a parametrization that is Cartesian. 36. The conoid is a special type of ruled surface given by σ (t, θ) = (r (t) cos θ, r (t)sinθ, z (θ)) = (0, 0,z(θ)) + r (t) (cos θ, sin θ, 0) Compute its first fundamental form. Show that if z (θ) =aθ for some constant a, thenr (t) can be reparametrized in such a way that we get a conformal parametrization. 37. Consider the two parametrized surfaces given by σ 1 (φ, u) = (sinhφ cos u, sinh φ sin u, u) = (0, 0,u)+sinhφ (cos u, sin u, 0) σ (t, θ) = (cosh t cos θ, cosh t sin θ, t) Compute the first fundamental forms for both surfaces and construct a local isometry from the first surface to the second. (The first surface is a ruled surface with a one-to-one parametrization called the helicoid, the second surface is a surface of revolution called the catenoid.) 38. Let S = x R 3 : x m = R.ShowthatSis a surface, and that if I and II denote the first and second fundamental forms, then II = ± 1 R I The conoid is a special type of ruled surface given by σ (t, θ) = (t cos θ, t sin θ, z (θ)) = (0, 0,z(θ)) + t (cos θ, sin θ, 0) Compute its first and second fundamental forms as well as the Gauss and mean curvatures. 5

6 40. Let γ (t) :I S be a regular curve on a surface S, with N being the normal to the surface. Show that κ n = II ( γ, γ) I( γ, γ),κ det ( γ, γ,n) g = (I ( γ, γ)) 3/ 41. Show that the principal curvatures at a point p S are equal if and only if at p the mean and Gauss curvatures are related by H = K. 4. Compute the matrix representation of the Weingarten map for a Monge patch σ (x, y) =(x, y, f (x, y)) with respect to the basis σ x, σ y. 6